cp's OEIS Frontend

Row sums => A000041. Diagonals are sums of Gaussian polynomials (which then sum to powers of two). The number of entries on each row is conjectured to conform to: 0 1 1 1 2 2 3 3 4 5 5 6 7 7 8 9 10 10 11 12 13 13 14 15 16 17 17 ... a sequence which stutters after values 0,1,2,4,6,9,12,16,...A002620.

Regarding the first element of the sequence as T(1,0), it appears that this is the number of partitions of n with k elements not in the first hook; i.e., with n - (max part size) - (number of parts) + 1 = k. If this is correct, we have T(n,0) = n and for k > 0, T(n,k) = sum_{j >= max(0,2k-n+2)} j * T(k,j). This is equivalent to T(n,k) = T(n-1,k) + sum_{j >= max(0,2k-n+2)} T(k,j) and thus to T(n,k) = 2* T(n-1,k) - T(n-2,k) + T(k,2k-n+2) [taking T(n,k) = 0 if k < 0]. It also implies the correctness of the conjecture about the row lengths. - Franklin T. Adams-Watters, May 27 2008

There are A033638(n) values in the n-th row, compliant with the order of the polynomial.

In the example for n=6 detailed below, the orders of [6, k]_q are 1, 6, 9, 10, 9, 6, 1 for k = 0..6,

the maximum order 10 defining the row length.

Note that 1 6 9 10 9 6 1 and related distributions are antidiagonals of A077028.

A083480 is a variation illustrating a relationship with numeric partitions, A000041.

The rows are formed by the nonzero entries of the columns of A049597.

The coefficient of q^j in the Gaussian polynomial [n, m]q is the number of binary words on alphabet {0,1} of length n having m 1's and j inversions. Hence T(n, k) is the number of length n binary words with exactly k inversions. - _Geoffrey Critzer, May 14 2017

If n is even the n-th row converges to n+1, n-1, n-4, ..., 19, 13, 7, 4, 3, 1 which is A029552 reversed, and if n is odd the sequence is twice A098613. - Michael Somos, Jun 25 2017

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).

The zeroth differences of a sequence are the sequence itself, while the k-th differences for k > 0 are the differences of the (k-1)-th differences. The distinct differences of any degree are the union of the k-th differences for all k >= 0. For example, the k-th differences of (1,1,2,4) for k = 0...3 are:

(1,1,2,4)

(0,1,2)

(1,1)

(0)

so there are a total of 4 distinct differences of any degree, namely {0,1,2,4}.

Row n of A049597 has a(n+1) nonzero values.

When considering the set of nested parabolas defined by -(x^2) + p*x for integer values of p, a(n) tells us how many parabolas are intersected by the line from (1,n) to (n,n). - Gregory R. Bryant, Apr 01 2013

Number of distinct perimeters for polyominoes with n square cells. - Wesley Prosser, Sep 06 2017

This triangle has the property that it contains the triangle A049597, since if we replace with zeros the positive terms before the first zero in the row n of this triangle, we get the triangle A049597.

Hence the sum of the terms after the last zero in row n equals A000041(n), the number of partitions of n (see the Example section).

Observation: at least the first 11 terms of column 1 coincide with A188674 (using the same indices).

Conjectured to be equal to A049597.

Begin with row zero and generate a column of values using the sequence of numeric partitions (A000041). At rows 1 4 9 16 25 ... A000290, generate two new columns one each to the left and right of existing columns. Note that the row sums appear to be A029552.